Values and tail values at risk of doubly compound inhomogeneous and contagious aggregate loss processes

This article provides efficient methods based on the saddlepoint approximation for computing the value at risk and the tail value at risk of the doubly compound and perturbed insurer total claim amount. The model is based on a primary counting birth process, for the number of catastrophic events, and on a secondary counting distribution, with possible modification or truncation at zero, for the number of individual losses generated from each catastrophe of the primary process. The considered primary processes are the inhomogeneous Poisson process and the homogeneous contagious Binomial and Negative Binomial processes, which have negative and positive contagions, respectively. The individual claim amounts are independent with a linear combination of Exponential distributions or with a Gamma distribution. The proposed methods are based on the saddlepoint approximation of Lugannani and Rice (1980) [12] and do not require Monte Carlo simulation. They are numerically very accurate, computationally efficient and hence relevant for the actuarial practice.